is \({}\geqslant 0\)  whenever defined, and moreover it is zero only for zero measure. This in turn implies (see e.g. [1]}) that all the measures in \(\mathfrak {M}\)  of finite \(\alpha \) -Riesz energy form a pre-Hilbert space \(\mathcal {E}_\alpha \)  with the inner product
\(\langle \mu ,\nu \rangle :=\kappa _\alpha (\mu ,\nu ):=\int \kappa _\alpha (x,y)\,d(\mu \otimes \nu )(x,y)\) 
